Introduction to finite fields

Fields and rings

To understand IDEA, AES, and some other modern cryptosystems, it is necessary to understand a bit about finite fields.

A field is an algebraic object. The elements of a field can be added and subtracted and multiplied and divided (except by 0). Often in undergraduate mathematics courses (e.g., calculus and linear algebra) the numbers that are used come from a field. The rational ⎧a ⎫ numbers Q =⎨:ab , are integers and b≠ 0⎬ form a field; fractions can be added (and ⎩⎭b subtracted) and multiplied (and divided). The real numbers R form a field. The complex numbers C form a field.

Number theory studies the integers Z . The integers do not form a field. Integers can be added (and subtracted) and multiplied, but integers cannot always be divided. Sure, 6 5 divided by 3 is 2; but 5 divided by 2 is not an integer; is a rational number. The 2 integers form a ring, but the rational numbers form a field.

Similarly the polynomials with integer coefficients form a ring. We can add (and subtract) polynomials with integer coefficients, and the result will be a polynomial with integer coefficients. We can multiply polynomials with integer coefficients, and the result will be a polynomial with integer coefficients. But, we cannot always divide X 2 − 4 XX3 + − 2 polynomials with integer coefficients: = X + 2 , but is not a X − 2 X 2 + 7 polynomial – it is a rational function. The polynomials with integer coefficients do not form a field, they form a ring. The rational functions with integer coefficients form a field.

A field has two operations; they are usually written as addition and multiplication. Subtraction is just the inverse of addition; it is adding the additive inverse (e.g., 5 – 4 = 5 + (-4) = 1). Division is just the inverse of multiplication; it is multiplying 1 by the multiplicative inverse (e.g., 6363÷ =×−1 =× 6 = 2). A ring also has two 3 operations – addition and multiplication – and, although addition is assumed to have an inverse, in a ring it is not assumed that multiplication has an inverse. (Addition and multiplication are also assumed to have several other properties. For both a ring and a field, it is assumed that addition commutes and is associative. For both a ring and a field, it is assumed that multiplication distributes over addition. For a field but not for a ring, multiplication is assumed to be commutative.)

The fields that we commonly used in mathematics courses ( Q , R , and C ) are infinite. For cryptological purposes, finite fields are useful.

Finite field of p elements

Recall that the integers mod 26 do not form a field. The integers modulo 26 can be added and subtracted, and they can be multiplied (so they do form a ring). But, recall that only 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, and 25 have multiplicative inverses mod 26; these are the only numbers by which we can divide. These twelve numbers are the positive integers that are less than or equal to 26 and relatively prime to 26.

Recall that given an integer that is less than or equal to n and relatively prime to n we can use the extended Euclidean algorithm to find its inverse mod n. So, it is possible to construct the multiplicative inverses of 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, and 25 modulo 26. If the modulus is prime, we can construct an inverse for each positive integer that is less than or equal to the modulus.. So, for each prime p we can construct a finite field of p elements – the integers mod p.

The integers mod 5 is a field of 5 elements {0, 1, 2, 3, 4}. Here are the addition and multiplication tables:

+ 01234 × 1234 001234 11234 112340 22413 223401 33142 334012 44321 440123

number additive inverse number multiplicative inverse 00 1 1 14 2 3

23 3 2 32 4 4 41

The identity for addition is 0, and the identity for multiplication is 1. This is a field. We

denote the field of 5 elements by F5

Here is F3 the finite field of 3 elements.

+ 01 2 × 12 0012 112 1120 221 2201

Here is the finite field of 2 elements F2 .

+ 01 × 1 001 11 110

Viewing 0 and 1 as bits, + is just XORing bits, and multiplication is … well, multiplication is not too interesting.

The American mathematician E.H. Moore (1862 – 1932) proved in 1893 that the number of elements of a finite field must be pn for some prime p and positive integer n, and he proved that for each prime p and positive integer n there is an essentially unique field of pn elements.